Question

Use the graph to answer the question.

The graph shows an exponentially rising line passing through (-2, 1.28) and (1, 2.5)

Write the exponential equation shown in the graph.

Answer 1

To write the exponential equation of the form \( y = ab^x \), we can use the two points given: (-2, 1.28) and (1, 2.5).

### Step 1: Set up the equations

Using the general form of the exponential equation \( y = ab^x \):

1. For the point (-2, 1.28):
\[
1.28 = ab^{-2} \quad \text{or} \quad 1.28 = \frac{a}{b^2} \quad \text{(Equation 1)}
\]

2. For the point (1, 2.5):
\[
2.5 = ab^1 \quad \text{or} \quad 2.5 = ab \quad \text{(Equation 2)}
\]

### Step 2: Solve the equations

From Equation 2, we can express \( a \) in terms of \( b \):
\[
a = \frac{2.5}{b} \quad \text{(substitute into Equation 1)}
\]

Now substitute \( a \) in Equation 1:
\[
1.28 = \frac{2.5}{b} \cdot \frac{1}{b^2}
\]
\[
1.28 = \frac{2.5}{b^3}
\]
Multiplying both sides by \( b^3 \) gives:
\[
1.28b^3 = 2.5
\]
Now solve for \( b^3 \):
\[
b^3 = \frac{2.5}{1.28} \approx 1.953125
\]
Taking the cube root:
\[
b \approx \sqrt[3]{1.953125} \approx 1.26
\]

Now substitute \( b \) back to find \( a \):
\[
a = \frac{2.5}{1.26} \approx 1.98
\]

### Step 3: Write the final equation

Now plug in the values of \( a \) and \( b \) into the exponential equation:
\[
y \approx 1.98 \cdot (1.26)^x
\]

Thus, the final exponential equation that represents the graph is approximately:
\[
y \approx 1.98 \cdot (1.26)^x
\]